The higher derivative theories are notorious for better convergency
properties at classical and quantum level, and a wider symmetry. In
most of the instances, these advantages come with a price of
dynamics instability, being the typical problem for the models in
this class. At the classical level the solutions to the equations of
motion demonstrate runaway behavior ("collapse"). At the quantum
level the ghost poles appear in the propagator, so the unitarity of
dynamics is an issue. These peculiarities follow from a single fact:
the canonical energy is unbounded in every non-singular higher
derivative theory. For a review of the problem, we cite recent
articles [1]}, [2]}, [3]} and references therein.
The canonical energy of singular models can be bounded. The examples
include \(f(R)\) -theories of gravity [4]}, [5]}, [6]}, [7]}. These models do not demonstrate instability. The
stability problem for constrained higher derivative theories is
discussed in [8]}. In the majority of interesting
(constrained or unconstrained) models, the instability has the
special form: the classical dynamics is regular (no "collapse"), but
the canonical energy is unbounded from below. The Pais-Uhlenbeck
oscillator [9]}, Podolsky [10]} and Lee-Wick [11]}, [12]} electrodynamics, ECS theory [13]}, and conformal gravity
[14]} are examples. For the current studies we mention
[15]}, [16]}, [17]}, [18]} and references therein. In all
these models the quantum theory is expected to be well-defined
[1]}, [20]}, but the application of the canonical
quantization procedure based on the Ostrogradski
procedureThe canonical Hamiltonian formalism for the
non-singular Higher-derivative theories has been first proposed in
[21]}. Its generalization for singular theories has been
developed in [22]}: see also [23]}. For recent studies see
[24]}, [25]}. leads to the model with an unbounded from below
spectrum of energy. That is why the stability and unitarity of
quantum dynamics is the most important issue.
